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Problem 79

Let T: \(\mathrm{R}^{3} \rightarrow \mathrm{R}^{2}\) be given by $\mathrm{T}\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}\right)=\left(7 \mathrm{x}_{1}+2 \mathrm{x}_{2}-3 \mathrm{x}_{3}, \mathrm{x}_{2}\right)$ As bases for \(\mathrm{R}^{3}\) and \(\mathrm{R}^{2}\) respectively, let $\mathrm{G}=\left\\{\mathrm{g}_{1}, \mathrm{~g}_{2}, \mathrm{~g}_{3}\right\\}=\\{(1,0,0),(0,1,-1),(0,0,1)\\}$ $\mathrm{H}=\left\\{\mathrm{h}_{1}, \mathrm{~h}_{2}\right\\}=\\{(1,0),(0,-1)\\}$

Expert verified

The matrix representation of the given linear transformation T with respect to bases G and H is:
\[T_G^H = \begin{pmatrix} 7 & 5 & -3 \\ 0 & -1 & 0 \end{pmatrix}\]

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Chapter 3

Let \(\mathrm{f}\) be the linear function from \(\mathrm{R}^{3}\) to \(\mathrm{R}^{2}\) such that \(\mathrm{f}(1,0,0)=(3,2), \mathrm{f}(0,1,0)=(1,4)\) and \(\mathrm{f}(0,0,1)=(2,-5)\) Find \(\mathrm{f}(2,0,5)\).

Chapter 3

Let \(\mathrm{T}: \mathrm{R}^{2} \rightarrow \mathrm{R}^{2}\) be given by $\mathrm{T}\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right)=\left(4 \mathrm{x}_{1}-2 \mathrm{x}_{2}, 2 \mathrm{x}_{1}+\mathrm{x}_{2}\right)$ and let \(\\{(1,1),(-1,0)\\}\) be a basis for \(\mathrm{R}^{2}\). Compute the matrix of \(\mathrm{T}\) in the given basis.

Chapter 3

Let \(\mathrm{T}\) be reflection about the \(\mathrm{y}\) -axis in \(\mathrm{R}^{2}\). Find the matrix of \(\mathrm{T}\).

Chapter 3

Let \(\mathrm{T}_{1}: \mathrm{R}^{4} \rightarrow \mathrm{R}^{3}\) and \(\mathrm{T}_{2}: \mathrm{R}^{4} \rightarrow \mathrm{R}^{3}\) be defined by $\mathrm{T}_{1}\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \mathrm{x}_{4}\right)=\left(6 \mathrm{x}_{1}+2 \mathrm{x}_{2}-3 \mathrm{x}_{3}, \mathrm{x}_{1}+4 \mathrm{x}_{3}+2 \mathrm{x}_{4}, 3 \mathrm{x}_{1}-\mathrm{x}_{2}+\right.$ \(\left.2 \mathrm{x}_{3}-5 \mathrm{x}_{4}\right)\) $\mathrm{T}_{2}\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \mathrm{x}_{4}\right)=\left(2 \mathrm{x}_{1}+4 \mathrm{x}_{2}-3 \mathrm{x}_{3}+5 \mathrm{x}_{4}, \mathrm{x}_{1}+2 \mathrm{x}_{2}+3 \mathrm{x}_{3}+\right.$ $\left.4 \mathrm{x}_{4}, 4 \mathrm{x}_{2}-2 \mathrm{x}_{3}-\mathrm{x}_{4}\right)$ Find, relative to the standard basis \(\\{(1,0,0,0),(0,1,0,0)\), \((0,0,1,0),(0,0,0,1)\\}\), the sum \(\mathrm{T}_{1}+\mathrm{T}_{2}\) and the linear combinations \(3 \mathrm{~T}_{1}+2 \mathrm{~T}_{2}\).

Chapter 3

Let \(\mathrm{T}\) on \(\mathrm{R}^{2}\) be given by \(\mathrm{T}(1,0)=3(1,0)+4(0,1)\) \(\mathrm{T}(0,1)=-5(1,0)+9(0,1)\) What is the matrix of \(\mathrm{T}\) with respect to \(\\{(1,0),(0,1)\\}\) ? b) Let the matrix \(\mathrm{A}\) be given by $$ \begin{array}{rl}\mathrm{A}= & 12 & 5 \\ & 14 & 9 & 3 \\ & & 1-1 & 8 & -5 & 1\end{array} $$ Find the linear operator \(\mathrm{T}\) on \(\mathrm{R}^{3}\) that \(\mathrm{A}\) represents, relative to \(\\{(1,0,0)(0,1,0)(0,0,1)\\}\)

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